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Doing Philosophy: From Common Curiosity to Logical Reasoning Timothy Williamson
The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle. Articles and Reviews 2006–2016 Michael Starks
The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John SearleArticles and Reviews 2006-2019 2nd Edition Michael Starks
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For those to whom this book is dedicated
Preface
I wrote this book because I like paradoxes … but even more than paradoxes, I like solutions. If you like such things too, then you might enjoy this book. I have offered my solutions to ten of what I found to be particularly fascinating and mind-boggling philosophical paradoxes. I hope that when you read the paradoxes, you feel puzzled and challenged, and that when you read the solutions, you feel a sense of things falling into place. In some cases, I hope you also come away with philosophically significant lessons.
I have written this book in such a way that, I hope, the generally educated reader can follow it. That is, although I assume you are generally smart and educated, I do not assume that you have read any of the literature on the paradoxes, nor any other specialized literature. Thus, I explain each paradox as if you don’t know what it is. When I want to address an idea that other philosophers have advanced, I explain that idea. I have made my explanations as clear and concise as I knew how to do, hoping neither to confuse you nor to waste your time. Complications and qualifications, as well as references to the literature, appear in footnotes.
At the same time, I have tried to write something of interest to professional philosophers. In many cases, my take on a paradox is distinctive and unorthodox. (If not for this, I would not have been motivated to write the book.) This is true particularly for chapters 2, 3, 6, 8, and 10.
Academic authors routinely overestimate their audiences – whatever our topic, we tend to vastly overestimate both the number of people interested in it, and the level of background knowledge people have about it. I think we overestimate the knowledge and interest even of other academics. (This involves a certain failure of reflection – we know, or should know, that our own knowledge of almost every other subject is minimal, yet we fail to consider that almost everyone else has about the same level of knowledge of our area of specialization.) Thus, there are many books that can only be usefully read by a handful of people in the world. I wrote this book in the way that I did because I wanted to make an intellectually valuable contribution, without adding to the stock of nearly-unread academic volumes. That is why it is as complex as it is, and why it is not more complex.
Now I would like to thank several other thinkers for discussion of some of the ideas in this book, including Iskra Fileva, Randall McCutcheon, Roy Sorensen, Eddy Chen, Christian Lee, Sam Director, David Barnett, Ari Armstrong, an anonymous referee for Palgrave, and the philosophers at the University of Vermont (to whom I presented chapter 3). Without their help, this book would be worse than it is. Naturally, none of these people can be blamed for any errors that remain. In the preface to my last book, I laid the blame for any errors on my PhD advisor, Peter Klein. I now realize that this was unfair and highly inappropriate. So I want to be clear that Peter is not to blame for the errors in this book. Of course, the blame for any errors falls entirely on my research assistant, Jasmine Carter, and my former student Matt Skene. I specifically instructed them to correct all errors in the manuscript. I thank them for their comments, and I will graciously accept their apology for my mistakes.
Fortunately, however, if the sentence you are now reading is true, then there are no errors in this book.
Boulder, CO, USA Michael Huemer
Analytical Contents
1 Introduction 1
1.1 What Is a Paradox? 1 Paradoxes are robust, widespread intellectual illusions in which seemingly compelling reasoning generates an absurd or contradictory conclusion.
1.2 What Is a Solution? 5
A solution should dispel the illusion, so that the paradoxical reasoning no longer seems compelling.
1.3 How to Seek Solutions 7 We should not expect a common approach to apply to all paradoxes – but self-evident principles such as those of classical logic must always apply.
1.4 Why Paradoxes? 10 Paradoxes are charming, fun, and may reveal deep confusions about important philosophical matters.
1.5 Paradoxes Not Covered 11 I address only philosophical paradoxes that do not depend on controversial views and that I have not previously addressed.
2.1 The Paradox 17
The liar sentence, “This sentence is false”, is apparently both true and false.
2.2 A Third Truth-Value 18
Some say the liar sentence is “indeterminate”. But what about the sentence, “This sentence is false or indeterminate”?
2.3 True Contradictions 19
The view that there are true contradictions is confused.
2.4 Meaninglessness 22
Perhaps the liar sentence is meaningless for one of the following reasons.
2.4.1 Self-Reference 23 Due to self-reference? But there are benign cases of self-reference.
2.4.2 False Presupposition 24 Due to containing a false presupposition? But we can easily remove the putative presupposition.
2.4.3 Lack of Communicative Use 25
Because it cannot be sincerely asserted? But other paradoxical sentences can be sincerely asserted. Because it cannot be used to convey information? But very similar sentences can be so used.
2.5 Putting the Blame on Truth 26
Some say that there is something wrong with the general concept of truth. But this approach is self-undermining and rules out too many innocent sentences.
2.6 A Solution 29
2.6.1 An Inconsistent Language 29 Our language contains inconsistent rules for how to interpret certain sentences, which results in sentences with no propositional content.
2.6.2 Meaning Deficiency 32
The liar sentence is not meaningless; it merely has a defective meaning that fails to pick out a proposition.
2.6.3 The Truth-Teller
33 “This sentence is true” also fails to express a proposition.
2.6.4 “The Liar Sentence Is Not True” Is True
34 It is permissible for a sentence other than L itself to say that L is not true.
2.6.5 This Sentence Is False or Meaning-Deficient
36 “This sentence is false or meaning-deficient” also fails to express a proposition.
2.6.6 Liar Cycles
In cases where a group of sentences generate a liar-like paradox, all the members of the group fail to express propositions.
2.6.7 Prohibiting Liars
36
38 There is no need to devise new syntactic rules for identifying liar-like sentences or ruling them “ungrammatical”.
2.7 Curry’s Paradox
Given a sentence, “This sentence is false, or p”, we can seemingly prove that p must be true. Again, the sentence fails to express a proposition.
2.8 The Paradox of Non-Self-Applicability
40
41 The property of “not applying to oneself” seemingly must apply to itself if and only if it does not apply to itself. Solution: there is no such property.
2.9 Russell’s Paradox
The set of all sets that don’t contain themselves must contain itself if and only if it doesn’t contain itself. Solution: there is no such set.
42
3.1
Removing a single grain from a heap of sand does not convert the heap to a non-heap. This principle entails that if a million grains of sand make a heap, then one grain of sand makes a heap.
3.2
3.3
Some respond with theories of indeterminacy or degrees of truth. These views have trouble explaining second-order vagueness. Degrees of truth introduce more precision than is plausibly present. Also, classical logic is self-evident.
Some say that a sentence is true provided that it would be true on any acceptable way of making the vague terms precise. This view has trouble with second-order vagueness. It also violates classical logic, conflicts with the T-schema, and implies that a statement of the theorist’s own view is false.
3.4 Epistemicism 55
Some say vague terms have precise boundaries that we merely fail to know. This is implausible since there is nothing that could make a particular boundary the correct one.
3.5
Mental states can be satisfied to varying degrees by different possible states of the world. The idea of the “propositional content” of a mental state is only a rough description of a mental state’s meaning, as if the state were always fully satisfied or fully unsatisfied.
3.5.2 When Thoughts Are Vague 63
Thoughts are vague when they have intermediate degrees of satisfaction in some possible situations. There are degrees of vagueness.
3.5.3 Uncertainty About Vagueness
We can be introspectively uncertain whether a thought is vague.
65
3.5.4 Vague Thoughts Make for Vague Language, and Vice Versa 66 The vagueness of thought makes language vague, and vice versa.
3.5.5 An Argument that Vague Statements Do Not Express Propositions 69 Vague sentences and thoughts do not express precise propositions, and there are no vague propositions, so vague sentences and thoughts don’t express propositions at all.
3.5.6 Arguments by Analogy
In other cases of semantic indecision, we accept that sentences fail to express propositions.
3.5.7 Logic Is Classical
72
74 This view of vagueness preserves the law of excluded middle and bivalence, properly understood.
3.5.8 How to Almost Say Something
76 Vague sentences often come close to expressing propositions.
3.5.9 Almost Is Good Enough 77 It is common and appropriate to use language approximately.
3.5.10 Applying Logic to Vague Sentences
79 We can apply our logical faculties to sentences even when they fail to express propositions.
3.5.11 Interpreting ‘Truth’: Strict Truth vs. Loose Truth
It is unsettled whether “is true”, as applied to sentences, means “expresses a true proposition” or instead means “expresses a thought that the world satisfies to a high degree”.
81
3.5.12 Why Is There Second-Order Vagueness? 84 First- and second-order vagueness both arise from the same features of mental states described above.
3.6 Conclusion 85
The sorites argument fails since none of its sentences express propositions. The premises almost express truths and the inference form is valid, but this does not guarantee a true or nearly true conclusion.
4 The Self-Torturer 91
4.1 The Paradox 91
The self-torturer repeatedly increases his torture level by undetectable increments, each time receiving a large financial reward. Seemingly rational individual choices lead to an intolerable end result.
4.2 Quinn’s Solution 93 Quinn holds that it is not always rational to choose the best option available at the time, and that rational choice is not always forward-looking.
4.3 An Orthodox Solution 95
4.3.1 In Defense of Undetectable Changes 95 The case actually shows that there can be unnoticeable changes in subjective experience.
4.3.2 Indeterminacy 97 It cannot be indeterminate how bad a pain is.
4.3.3 In Defense of an Optimal Setting 99 Since pain has constant marginal disutility, while money has diminishing marginal utility, there is an optimal point for the self-torturer to stop.
4.3.4 Detectable and Undetectable Values
103 It is not so strange that an undetectable bad can outweigh a detectable good. Undetectable quantities can often be larger than detectable ones.
4.3.5 Advantages of This Solution 105 My solution to the problem preserves classical logic and decision theory, without positing anything particularly strange.
5.1 The Paradox
You are asked to choose between taking box A and taking both A and B, where B contains $1000, and A contains either $1 million (if a reliable predictor thought you would take only A) or $0 (if the predictor thought you would take both). Dominance reasoning supports taking both, but expected utility maximization seemingly supports taking only A.
5.2 Objections to the Scenario 110 The scenario can be made more realistic without eliminating the paradox.
5.3 The Right Expected Utility Principle
5.3.1 The Right Way to Make Good Things More Likely
Rational agents seek to “make more likely” the achievement of their goals in a causal sense, not an evidential sense.
5.3.2 Two-boxing Maximizes Expected Utility: Doing the Math
115 When we incorporate the above insight into the notion of expected utility, Dominance and Expected Utility principles agree.
5.3.3 Why This Is the Best Solution
119 The causal decision theorist’s solution preserves central intuitions about rational choice.
5.4 The Case of Perfect Reliability
119 What if the predictor is 100% reliable? This requires either backward causation (making one-boxing rational) or determinism (ruling out free choice).
5.5 Rationality and Long-Run Benefit
One-boxers say that rationality must be tied to how much one benefits in the long run. They then appeal to one of the following claims.
122
5.5.1 One-Boxers as a Group Do Better
122 But the wellbeing of the group one belongs to is irrelevant in rational choice theory.
5.5.2 One-Boxers Tend to Do Better in any Given Case 124 But this rests on a mistaken interpretation of the probabilities used in rational choice theory.
5.5.3 One-Boxers Do Better in Repeated Games 124 But this changes the scenario in a way that may also change the causal decision theorist’s answer.
5.5.4 Being a One-Boxer Is Predictably Advantageous
125 But this only shows that there are scenarios that systematically reward irrationality. This applies to any paradigmatic form of irrationality.
5.6 Uncertainty About Decision Theory
128 In decision-making, it can be rational to give some weight to each of two conflicting theories of rational choice.
6 The Surprise Quiz
6.1 The Paradox 133
The teacher announces a surprise quiz next week. It can seemingly be shown that there is no day on which the quiz can occur.
6.2 Rejecting the Assumptions
The scenario uses idealized assumptions about the students’ reasoning. But these assumptions can be relaxed without detriment to the paradox.
6.3 What Is Surprise? 136 We may assume that there is a threshold level of antecedent credence in an event that renders the event non-surprising.
6.4 Quiz Comes if and only if Surprising
In one version of the story, we assume that the professor has no interest in giving a quiz unless it will be a surprise.
6.4.1 Self-Undermining Beliefs with a Vague Surprise Threshold
If the quiz has not come by Thursday, the students should adopt a credence that would make a Friday quiz a borderline case of a surprise.
6.4.2 Self-Undermining Beliefs with a Precise Threshold
138
139
140 Or they should adopt a credence such that they will merely not know whether the Friday quiz would count as a surprise.
6.4.3 The Rest of the Week
142 Thursday and earlier quizzes will then count as surprising.
6.5 Quiz Comes, with or without Surprise
143 In another version of the story, we assume the professor will give a quiz whether or not it will surprise.
6.5.1 No Friday Surprise
144 In this version, a Friday quiz would not surprise.
6.5.2 Borderline Thursday Surprise
A quiz on Thursday would be a borderline case of a surprise, or it would be unknown whether it counted as a surprise.
6.5.3 The Rest of the Week
A quiz on any earlier day would be a surprise.
6.6 Surprising as Not-Most-Expected
If a quiz counts as “surprising” only when it was not antecedently considered most likely to occur on that day, then a surprise quiz cannot be given.
144
145
146
7
7.1
There are two indistinguishable envelopes containing money, one with twice as much as the other. It can be argued that each envelope has a higher expected value than the other.
7.2 The Use of Probability in the Paradox 151
7.2.1 An Objection 151 Some think the paradoxical reasoning misuses the concept of probability.
7.2.2 Three Interpretations of Probability 152 Probability can be interpreted epistemically, subjectively, or physically.
7.2.3 Rational Choice Uses Epistemic Probabilities 153 Epistemic probability is the correct interpretation for rational choice theory. The paradox thus does not misuse the concept of probability.
7.2.4 Probabilities in Causal Decision Theory 154 This does not conflict with our earlier defense of causal decision theory.
7.3 The Use of Variables in the Paradox 155 The paradoxical reasoning confuses variables with constants.
7.4 The Correct Analysis
A correct analysis would assign a coherent probability distribution to each possible way of distributing money across the two envelopes. This leads to both envelopes having the same expected value.
8 The Principle of Indifference
The PI holds that, given no reason for preferring any of a set of alternatives over any other, all are equally probable.
8.2 The Paradoxes of the Principle of Indifference
There are cases in which the PI can seemingly be used to justify incompatible probability assignments.
8.2.1 The Colored Book
Given that a book is red, green, or blue, what is the probability that it is red?
162
162
8.2.2 France and England 163
Given that England is a proper part of the U.K., what is the probability that England is smaller than France? That the U.K. is smaller than France?
8.2.3 The Car Ride
Given a 100-mile car trip with duration between 1 and 2 hours, what is the probability of a duration between 1 and 1.5 hours? What about a velocity between 66.7 and 100 mph?
163
8.2.4 The Cube Factory 165
Given that a cube is between 0 and 2 inches on a side, what is the probability of a side between 0 and 1 inch? What about a volume between 0 and 1 cubic inch?
8.2.5 The Circle and Chord 165
Given a circle with an equilateral triangle inscribed inside, what is the probability of a random chord being longer than a side of the triangle?
8.3 Wherefore Indifference?
168
8.3.1 Theories Rejecting the PI 168
Empiricists say we can have no probabilities prior to empirical evidence. Subjectivists say any coherent initial probabilities are permissible. These views reject the PI.
8.3.2 The PI Is Intuitive
169 The PI is extremely intuitive in many cases.
8.3.3 The PI Is an Analytic Truth
170 The PI seems to follow from the meaning of epistemic probability. Without a priori probabilities, there are no epistemic probabilities at all.
8.3.4 The PI Underlies the Least Controversial Probability Assessments 172 Probability assessments based on randomization or statistical evidence also depend on the PI.
8.4 Interpreting the Principle of Indifference: The Explanatory Priority Proviso 174 The PI should be applied to the most explanatorily basic set of possibilities.
8.5 Solutions
175
8.5.1 The Colored Book 175 A uniform probability density should be assigned over the color solid.
8.5.2 France and England
177 Equal probabilities should be assigned to each possible complete set of “larger than” relations involving England, France, and the U.K.
8.5.3 The Car Ride 178 Velocity is privileged over time since velocity causally determines duration, given a fixed distance.
8.5.4 The Cube Factory 179 Quantity of material is privileged over width or volume since quantity of material causally determines size.
8.5.5 The Circle and the Chord
180 The PI should be applied by averaging over all known methods of random selection of a chord.
8.6 A Philosophical Application: The Problem of Induction 183
8.6.1 The Traditional Problem 183 Skeptics say that there is no reason to believe the course of nature is uniform, and hence no justification for relying on induction.
8.6.2 A Probabilistic Formulation of the Problem 184 Skepticism can be defended by applying the PI to all possible sequences of observations.
8.6.3 A Solution 186 We should instead apply the PI to all possible values of the objective chance of a given event.
8.6.4 The Mathematics of the Inductivist Solution 188 This can be used to derive Laplace’s Rule of Succession.
8.7 Another Application: The Mystery of Entropy 189
8.7.1 Why Entropy Increases 189 There is a probabilistic argument for why, starting from a low-entropy state, the entropy of a physical system should spontaneously increase.
8.7.2 The Reverse Entropy Law 191 A time-reversed version of that argument can be used to conclude that any given lowentropy state was probably preceded by a higher-entropy state.
8.7.3 Reverse Entropy Is Crazy 193 This has all sorts of ridiculous implications.
8.7.4 The Reverse Argument Misuses the Principle of Indifference 196 The argument for Reverse Entropy seems to ignore that the past is explanatorily prior to the present.
8.7.5 The Isolated Box 199 The diagnosis of the preceding subsection may prove too much. It is only plausible in a certain range of cases.
9 The Ravens
9.1 A Paradox of Confirmation
It seems that, in general, the observation of an A that is B supports “All A’s are B.” Therefore, observation of a purple shoe supports “All non-black things are non-ravens.” This is logically equivalent to “All ravens are black.” So purple shoes provide evidence that all ravens are black.
9.2 Solution
Whether an observation of a purple shoe supports “All ravens are black” or not depends upon how the observation was gathered – e.g., whether it was gathered by selecting randomly from the class of non-ravens, or selecting randomly from the class of non-black things.
10
The Shooting Room is set up such that (i) it is guaranteed that at least 90% of people who ever enter it are shot, but (ii) for any given person, whether they are shot or not depends on the flip of a fair coin. Q: Given that V is called into the room, what is the probability that V is shot?
In any finite version of the shooting room scenario, the correct probability comes to 50%.
10.2.2 The Impossibility of the Infinite Case
The paradox depends on metaphysically impossible assumptions about an infinite population of potential victims, and infinite speed or elapsed time.
Beauty is put to sleep and woken up either once or twice, depending on the flip of a coin; after each
waking, she will fall asleep and forget having woken. Upon waking, what should be her credence that the coin came up heads? Some say 1/2; others say 1/3.
11.2 The Fine Tuning Argument 221
The apparent fine tuning of the universe’s physical parameters, which is required for life to exist, might be evidence that there are many universes.
11.3 The Doomsday Argument 224
The number of people who have existed before you might constitute evidence that not many more will exist after you.
11.4 The Multiverse: Pro and Con 227
11.4.1 The “This Universe” Objection 227 Some object that the fine tuning evidence does not support multiple universes, because the existence of other universes would not explain anything about this universe.
11.4.2 In Defense of the Multiverse 229
The “this universe” objection seems parallel to some incorrect objections in other cases.
11.4.3 Four Cases Resolved 231
Evidence supports a theory (for you) when your having that qualitative evidence would be more likely if the theory were true than if it were false. This view gives the right verdicts in four cases of interest.
11.4.4 Personal Identity and the Multiverse Theory 234
Conclusion: the fine tuning evidence supports multiple universes, if and only if it would be possible for you to exist in another universe.
11.5 Against Doomsday 236
The Doomsday Argument fails since, given the impossibility of backward causation, no hypothesis about humanity’s future affects the probability of your now having the evidence that you have.
11.6 Sleeping Beauty: For a Third 239 Upon waking, Beauty should update on the evidence, “I am awake now”. This results in a credence of 1/3 that the coin came up heads.
12 Concluding Remarks
12.1 Seven Varieties of Error
The preceding paradoxes exhibit several kinds of problem that also beset human thinking in more ordinary cases. These include: hidden assumptions, neglect of the small, confusion, binary thinking, oversimplification, inappropriate idealization, and inference from partial data.
12.2 Against Radical Revision
245
245
251 We should not give up extremely obvious principles, such as those of classical logic, to avoid paradox. We should prefer to qualify principles rather than rejecting them outright. Our mistakes are likely to be subtle, not blatant.
12.3 Reality Is Intelligible, with Difficulty 252 The world is not inconsistent or incomprehensible. Human reason is highly fallible but correctable with effort.
List of Figures
1.1
Fig. 2.1 Classical concept of negation
Fig. 2.2 A non-classical conception of negation
Fig. 3.1 Satisfaction profile of a desire
Fig. 3.2 Simplified desire profile
Fig. 3.3 Profile of belief that H is a heap
Fig. 3.4 Degrees of vagueness 77
Fig. 3.5 A vague belief 85
Fig. 4.1 Optimal stopping point for the self-torturer 101
Fig. 4.2 A case with constant disvalue of pain and no optimal stopping point 101
Fig. 4.3 A case with diminishing disvalue of pain and no optimal stopping point 102
Fig. 6.1 Optimal credence with a vague surprise threshold 140
Fig. 6.2 Credences with a precise surprise threshold 141
Fig. 8.4 Migrant from Indonesia to the Netherlands 199
Fig. 9.1 Ways of collecting an observation of a non-black non-raven 205
Fig. 11.1 Sleeping Beauty problem 220
Fig. 11.2 The multiverse 223
1
Introduction
1.1 What Is a Paradox?
First, some words about what a paradox isn’t. Some people understand the word “paradox” to refer to a case in which reality is contradictory, that is, a situation that you would correctly describe by contradicting yourself. I do not use the word this way, because I find it inconvenient. If we use “paradox” to denote a situation containing a true contradiction, then we will have to say that, by definition, there are no paradoxes, since contradictions are necessarily false – thus apparently depriving this book of its subject matter. I should then have to say that this is a book about “apparent paradoxes”, that the next chapter is about “the Liar Pseudo-paradox”, and so on. This would be tedious. So I won’t understand “paradox” that way. Some people use “paradox” simply to refer to a contradictory statement, or apparently contradictory statement, such as “Nobody goes to that restaurant anymore because it is too crowded.” That also is not what I mean by “paradox”. Such statements are either false, or simply have an alternate meaning that is different from the most superficial interpretation (as in the statement, “I am nobody”, which really just means “I am unimportant”). In either case, there is no real puzzle.
M. Huemer, Paradox Lost, https://doi.org/10.1007/978-3-319-90490-0_1
M. Huemer
I understand a paradox, roughly, as a situation in which we have seemingly compelling reasoning for a contradictory or otherwise absurd conclusion.1 We feel that we cannot accept the conclusion, but nor can we readily identify a flaw in the reasoning. For example, consider Zeno’s famous paradox of motion (figure 1.1):
In order for an object to move from point A to point B, the object must first travel half the distance. Then it will have to travel half the remaining distance. Then half the remaining distance again. And so on. This is an infinite series. An infinite series has no end; hence, it is impossible to complete an infinite series. Therefore, it is impossible for the object to reach point B. Thus, no object can move anywhere.
The conclusion is absurd, so the reasoning must be wrong. Nevertheless, the reasoning has a certain obvious, intuitive force, and it is difficult to say exactly what is wrong with it.
To count as “paradoxical”, the reasoning for the absurd conclusion must have widespread appeal – that is, the reasoning must be of a sort that would seem compelling to typical human reasoners; an idiosyncratic error that I personally can’t seem to shake does not qualify. Thus, if I make a calculation error in multiplying a string of numbers, which results in my deriving a logically impossible conclusion, this will not count as a paradox – not even if I personally cannot find the error after many tries.
To count as “paradoxical”, a piece of erroneous reasoning must also have a certain sort of robustness: paradoxes bear extended contemplation and discussion. Paradoxes can have solutions and attempted solutions, but the correctness of a given solution will be a matter of debate, at least for some
1 Similarly, Sainsbury (2009, p. 1) defines a paradox as “an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises”. But see below in the text for further conditions on paradoxicality.
Fig. 1.1 Zeno’s paradox
time, even among the experts. A paradox is not merely a problem whose solution, though known to experts, is unknown to most non-experts.
Thus, for example, I do not consider the Monty Hall Problem to be a paradox. The Monty Hall Problem goes like this:
You are a contestant on the game show Let’s Make a Deal, with host Monty Hall. You know how the game works: at a certain point in the game, Monty shows the contestant three closed doors. One of the doors has a nice prize behind it (say, a new car); the other two have goats behind them (assume that no one wants a goat). The contestant is allowed to choose one of the doors, and will be allowed to have whatever is behind it. After the contestant chooses, but before he reveals what is behind the chosen door, Monty opens one of the other two doors and shows the contestant a goat.2 He never opens the door with the car behind it; he always shows the contestant a goat.3 Monty then asks if the contestant would like to change their choice, that is, to switch to the other closed door. Thus, suppose you initially
2 This is not exactly how the real game worked, but pretend the game works this way for purposes of the problem. In the real game, Monty was not required to show the contestant a goat or offer the chance to switch, and usually he did not do so (Tierney 1991).
3 The stipulation that Monty always opens a door with a goat behind it is sometimes erroneously omitted from the statement of the problem, as in vos Savant (1990–91) (vos Savant makes the assumption in her solution, but the original problem statement did not contain it). Without this stipulation, the correct probability is ½. That is, suppose we assume that Monty, rather than deliberately avoiding the door with the prize, simply chooses randomly which door to open, from the two doors that the contestant didn’t pick. Let h1 = [The car is behind door 1], h2 = [The car is behind door 2], h3 = [The car is behind door 3], and e = [Monty opens door 3 and there is a goat behind it]. After you have chosen door 1 but before Monty opens door 3, you should have the following credences: P(h1) = P(h2) = P(h3) = ⅓; P(e|h1) = ½; P(e|h2) = ½; P(e|h3) = 0. Then the probability of door 1 having the prize behind it, given that Monty opens door 3 and reveals a goat, is given by Bayes’ Theorem as follows:
The key is that in this version of the problem, P(e|h2) = ½. In the standard version (where Monty always avoids opening the door with the prize), P(e|h2) = 1. Substituting 1 for P(e|h2) in the above equation changes the final answer to ⅓, the standard answer. Introduction
M. Huemer
choose door #1. Monty then opens, say, door #3 and shows you a goat behind it. He then asks if you would like to change your choice from door #1 to door #2. Should you switch?
Most people have a strong intuition that it doesn’t matter whether you switch to door #2 or stick with door #1; that’s because most people think that the prize is now 50% likely to be behind door #1 and 50% likely to be behind door #2. The correct answer, however, is that you should definitely switch to door #2: door #1 has a 1/3 probability of having the real prize behind it, and door #2 now (after you saw the goat behind door #3) has a 2/3 probability of having the real prize.
It can be difficult to convince people of this. In fact, almost everyone, on first hearing the problem, gives the wrong answer, and persists in that answer until bludgeoned for a while with probability calculations or experiments.4 In this case, there are compelling arguments (discussed below and in fn. 3) for a highly counter-intuitive answer. Nevertheless, I do not consider this a paradox. One reason is that this problem is not robust enough to bear debate among experts. The Monty Hall Problem has a well-known, objectively correct solution that can be shown to be so in a fairly brief span of time; it does not, for example, bear years of reflection.
Why does the prize have a 2/3 probability of being behind door B? This is beside my present point (which, remember, was just to define “paradox”); however, in case you can’t sleep until you know, an explanation follows. (You can also do a calculation employing Bayes’ Theorem, but the following is probably going to be more satisfying.)
Suppose Monty runs the game 300 times. Each time, the location of the good prize is randomly selected from among the three doors. We would expect that in about 100 of these games, the contestant’s initial guess is correct, that is, the first door they pick has the prize behind it. The other 200 times, the initial guess is wrong. Therefore, if the contestants always stick with their initial guess, then 100 of the 300 will win the real prize, and 200 will receive goats. (The 200 who initially selected a goat door can’t possibly improve their result by sticking with that choice!)
4 See the responses to Marilyn vos Savant’s famous column on the problem (vos Savant 1990–91).
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elapsed, when the respiration became slow, and it ceased at two minutes and three-quarters after the mouse was introduced. It was removed at this time, and it gasped a few seconds afterwards; this gasp was soon followed by another; the gaspings became more frequent, and in a short time, the natural breathing was resumed. In five minutes, the mouse was able to walk.
The third, fourth, and fifth of the above experiments show that one grain of chloroform to each hundred cubic inches of air suffices to induce the second degree of narcotism, or that state in which consciousness and voluntary motion are disturbed, but not entirely abolished. Now one grain of chloroform produces 0·767 of a cubic inch of vapour at 60°, when its specific gravity is 4·2; and, when the vapour is inhaled, it expands somewhat, as it is warmed to the temperature of the lungs; but it expands only to the same extent as the air with which it is mixed, and therefore the proportions remain unaltered. But air, when saturated with vapour of chloroform at 100°, contains 43·3 cubic inches in 100; and
As 0·767 : 43·3 :: 0·0177 : 1.
So that if the point of complete saturation be considered as unity, 0·0177 or 1–56th, will express the degree of saturation of the air from which the vapour is immediately absorbed into the blood; and, consequently, also the degree of saturation of the blood itself.
I find that serum of blood at 100°, and at the ordinary pressure of the atmosphere, will dissolve about its own volume of vapour of chloroform; and since chloroform of specific gravity 1·483 is 288 times as heavy as its own vapour, 0·0177 ÷ 288 gives 0·0000614, or one part in 16,285, as the average proportion of chloroform by measure in the blood, in the second degree of narcotism.
It is evident, from the experiments numbered 9 to 12 inclusive, that two grains of chloroform to each hundred cubic inches of the inspired air cause a state of very complete insensibility, corresponding with what I have designated the fourth degree of narcotism; and by the method of calculation employed above we get 0·0354, or 1–28th, as representing the degree of saturation of the blood, and 0·0001228 the proportion by measure in the blood.
In experiments 6, 7, and 8, in which quantities of chloroform were employed intermediate between one and two grains to each hundred
cubic inches of air, a moderate amount of insensibility was induced, corresponding very much with the state of patients during operations under chloroform.
The experiments from 13 to 18 show that quantities of chloroform, exceeding two grains to 100 cubic inches of air, have a tendency to embarrass and arrest the function of respiration, if the inhalation is continued. I have not yet been able to determine satisfactorily the exact proportion of chloroform which requires to be absorbed to arrest the respiration of animals of warm blood. I believe there is a definite proportion which has this effect, but there are two reasons why it is not so easy to ascertain it, as to ascertain the proportion which causes the minor degrees of narcotism. In the first place, the breathing often becomes very feeble before it ceases, so that the animal inhales and absorbs but very little chloroform, and remains on the brink of dying for some time. In the next place, the temperature of the body falls in a deep state of narcotism, especially in small animals; and, as the temperature falls, the amount of chloroform which the blood can dissolve from any given mixture of air and vapour increases.
Judging from the experiments numbered 14 to 18, three grains of chloroform to each hundred cubic inches of air must be very nearly the quantity which has the power of arresting the breathing when the temperature of the body is 100°; and as three grains of chloroform produce 2·3 cubic inches of vapour, and air at 100° is capable of taking up 43·3 per cent. of its volume, it follows that the blood must contain between 1–18th and 1–19th as much chloroform as it is capable of dissolving, at the time when the respiration is arrested. In the 14th experiment, the breathing of the two mice was on the point of being stopped by two and a half grains of chloroform in each hundred cubic inches of air, but during the thirteen minutes which the mice breathed the vapour, their temperature fell to about 90°. Air, when saturated with the vapour of chloroform at this temperature, contains 35 per cent., and two grains and a half of chloroform yield 1·917 cubic inches of vapour; so by a calculation similar to that made at page 68, the mice at the time when the breathing was about to cease must have absorbed 1–18th part as much chloroform as their circulating fluids were capable of dissolving.
The reader will have observed that, in the experiments related above, the mice became much more quickly affected than the guineapigs and cats. The reason of this is their quicker respiration and circulation, and much more diminutive size. Little birds, such as linnets and sparrows, are also very quickly affected by chloroform. Frogs are more slowly affected, owing to their languid respiration, unless the vapour to which they are exposed is very strong.
They can, however, owing to their low temperature, be rendered insensible by proportions of vapour too small to affect animals of warm blood; and as they have no proper temperature of their own, the amount of vapour (in proportion to the air in which they are placed) that will affect them, depends entirely on the temperature of that air.
The following experiment was several times performed on frogs with the same result, the temperature of the room being about 55°, as it was in winter.
Experiment 19. 4·6 grains of chloroform were diffused through the air of a jar of the capacity of 920 cubic inches, and a frog was introduced. In a few minutes, it became affected, and at the end of ten minutes, was quite motionless and flaccid; but the respiration was still going on. Being now taken out, it was found to be insensible to pricking: it recovered in a quarter of an hour.
In a repetition of this experiment, in which the frog continued a few minutes longer in the vapour, the respiration ceased, and the recovery was more tardy. On one occasion, the frog was left in the jar for an hour, but when taken out, and turned on its back, the pulsations of the heart could be seen. In an hour after its removal, it was found to be completely recovered.
The first of the experiments related above (page 60), showed that an atmosphere containing half a grain of chloroform to each hundred cubic inches, produced scarcely any appreciable effect on animals of warm blood; but the following calculation explains why this quantity acts so energetically on the frog, and proves that this creature is affected by chloroform according to the same law as animals of warm blood. The vapour is absorbed into the blood and lymph of the frog at the temperature of the external air, whose point of relative saturation therefore remains unaltered, both in the lungs and in contact with the skin of the animal; and as half a grain of chloroform
produces 0·383 cubic inches of vapour, and air at 55° contains, when saturated, 10 per cent. of vapour; 0·0383, or 1–26th, expresses the degree of saturation of the air, and also of the blood of the frog. And this is a very little more than the quantity (0·0354 or 1–28th) which was calculated above to be the greatest amount which could be absorbed with safety into the blood of the mammalia. It must be observed, however, that the pulmonary respiration of the frog was arrested by this proportion of 1–26th as much chloroform as the blood would dissolve, whilst we calculated that it required about as much as 1–18th to arrest the breathing of animals of warm blood. It must be remembered, however, that the pulmonary respiration of frogs is a process of swallowing air, which only goes on when the creature is comparatively active. In the torpid state, the respiration takes place only by the skin, and the frog never breathes with the aid of the same muscles and nerves as mammalia and birds.
By warming a frog, together with the air in which it is placed, it is, in accordance with the law explained above, rendered comparatively proof against an amount of chloroform which would otherwise render it insensible.
Experiment 20. A frog, which had been a few days previously subjected to the experiment just narrated, was put into the same jar, which was placed near the fire, till a thermometer inside marked 75° Fah.; 4·6 grains of chloroform were then introduced, and diffused through the air in the jar. The jar was kept for twenty minutes, with the thermometer indicating the same temperature within one degree. For the first seventeen minutes, the frog was unaffected; and only was dull and sluggish, but not insensible, when taken out. Air at 75°, when saturated with vapour of chloroform, contains 22 per cent., and therefore the 0·383 per cent. of vapour, which at 55° was capable of saturating the fluids of the frog to the extent of 1–26th of what they would dissolve, was, at 75°, capable of saturating them only to the extent of 1–57th.
At one of Dr. Wilson’s Lumleian Lectures, at the College of Physicians, on March 29th, 1848, I had the honour of performing some experiments, and making some remarks, on chloroform, and I combined together two experiments on frogs and small birds, in a way which shows how entirely the effects of a narcotic vapour
depend on the quantity of air with which it is mixed, and on other physical conditions.
Experiment 21. I introduced a chaffinch, in a very small cage, into a glass jar holding nearly 1,000 cubic inches, and put a frog into the same jar, covered it with a plate of glass, and dropped five grains of chloroform on a piece of blotting paper suspended within. In less than ten minutes, the frog was insensible, but the bird was not affected.
Experiment 22. I then placed another frog and another small bird in a jar containing but 200 cubic inches, with exactly the same quantity of chloroform. In about a minute and a half, they were both taken out,—the bird totally insensible, but the frog not appreciably affected, as from its less active respiration it had not had time to absorb much of the vapour.
The blood in the human adult is estimated by M. Valentin to average about thirty pounds. M. Valentin’s experiments were so conducted that this quantity must include the extra vascular liquor sanguinis, as well as the blood actually contained within the vessels. On this account, his estimate is all the better fitted for calculating the amount of chloroform absorbed, since this medicine, when inhaled gradually, passes by exosmosis through the coats of the bloodvessels into the fluid in which the tissues are immediately bathed. The above quantity of blood would contain 26 pounds 5 ounces of serum, which, allowing for its specific gravity, would measure 410 fluid ounces. This being reduced to minims, and multiplied by 0·0000614, the proportion of chloroform in the blood required to produce narcotism to the second degree (see page 68), gives 12 minims as the whole quantity in the blood. More than this is used in practice, because a considerable portion is not absorbed, being thrown out again when it has proceeded no further than the trachea, the mouth and nostrils, or even the face-piece. But I find that if I put twelve minims into a bladder containing a little air, and breathe it over and over again, in the manner of taking nitrous oxide, it suffices to remove consciousness, producing the second degree of its effects.
To induce the third degree of narcotism, or the condition in which surgical operations are usually commenced, would require that about 18 minims should be absorbed by an adult of average size and health, according to the above method of calculation; and to induce the deep
state of insensibility, which I have termed the fourth degree of narcotism, would require 24 minims; whilst to arrest the function of respiration would require that about 36 minims should be absorbed.
PREPARATIONS FOR INHALING CHLOROFORM.
The only direction which it is usually requisite to give beforehand, to the patient who is to inhale chloroform, is to avoid taking a meal previous to the inhalation; for chloroform is very apt to cause vomiting, if inhaled whilst there is a quantity of food in the stomach. The sickness is not attended with any danger, but it constitutes an unpleasantness and inconvenience which it is desirable to avoid. The best time of all for an operation under chloroform is before breakfast, but the customs and arrangements of this country do not often admit of that time being chosen, and it is unadvisable to make the patient fast beyond his usual hour. It answers very well to perform an operation about the time when the patient would be ready for another meal, or, if the time of operation fall two or three hours after the usual time of eating, to request the patient to make only a slender repast at that time, so as just to prevent the feeling of hunger. It is impossible to prevent vomiting in some cases with the best precautions, for the stomach occasionally will not digest when the patient is expecting a surgical operation, and the breakfast may be rejected in an unaltered state hours after it has been taken. In other cases the patient does not vomit, even when he inhales chloroform shortly after a full meal.
The most convenient position in which the patient can be placed whilst taking chloroform is lying on the back, or side, as he is then duly supported in the state of insensibility, and can be more easily restrained if he struggle whilst becoming insensible. The semirecumbent posture on a sofa does very well, and there is no objection to the sitting posture, when that is most convenient to the operator. In that case, however, the patient should be placed in a large easy chair with a high back, so that the head as well as the trunk may be supported without any effort, otherwise he would have a tendency to slide or fall when insensible. It has been said that it is unsafe to give chloroform in the sitting posture, on the supposition that it would in some cases so weaken the power of the heart, as to render it unable
to send the blood to the brain. Observation has proved, however, that chloroform usually increases the force of the circulation; and although the horizontal position is certainly the best for the patient under an operation in all circumstances, I consider that the sitting posture is by no means a source of danger, when chloroform is given, if the ordinary precaution be used, which would be used without chloroform—that of placing the patient horizontally if symptoms of faintness come on. I have preserved notes of nine hundred and fortynine cases in which I have given chloroform to patients in the sitting posture, and no ill effects have arisen in any of these cases.
The person who is about to inhale chloroform is occasionally in a state of alarm, either about that agent itself or the operation which calls for its use. It is desirable to allay the patient’s fears, if possible, before he begins to inhale, as he will then be able to breathe in a more regular and tranquil manner. In a few cases, however, the apprehensions of the patient cannot be removed, and they subside only as he becomes unconscious from the inhalation. It has been said that chloroform ought not to be administered if the patient is very much afraid, on the supposition that fear makes the chloroform dangerous. This is, however, a mistake; the danger, if any, lies in the fear itself. Two cases will be related hereafter in which the patients died suddenly from fear, whilst they were beginning to inhale chloroform, and before they were affected by it; but the probability is that, if they had lived till the chloroform took effect, they would have been as safe as other patients who inhaled it. If chloroform were denied to the patients who are much afraid, the nervous and feeble, who most require it, would often be deprived of its benefits. Moreover, the patients would either be prevented altogether from having the advantage of surgery, or they would be subjected to the still greater fear of the pain, as well as the pain itself; for whatever undefined and unreasoning fears a patient may have when the moment comes for inhaling chloroform, he has only chosen to inhale it on account of a still greater fear of pain.
Fear and chloroform are each of them capable of causing death, just as infancy and old age both predispose to bronchitis, but it seems impossible that fear should combine with the effects of chloroform to cause danger, when that agent is administered with the usual precautions. Fear is an affection of the mind, and can no
longer exist when the patient is unconscious; but the action of that amount of chloroform which is consistent even with disordered consciousness is stimulating, and increases the force and frequency of the pulse, in the same way as alcohol. I believe that no one would assert that a person would die the sooner of fright for having taken a few glasses of wine, or a small amount of distilled spirits, whatever might be the state of his health. When chloroform has been absorbed in sufficient quantity to cause unconsciousness, fear subsides, and with the fear its effects on the circulation. It is a subject of almost daily observation with me that the pulse, which is extremely rapid from some ill defined apprehension, when certain patients begin to inhale chloroform, settles down to its natural frequency after they become unconscious.
The practice I have always followed has been to try to calm the patient, by the assurance that there was nothing to apprehend from the chloroform, and that it would be sure to prevent all pain; but where it has been impossible to remove the fears of the patient in this way, I have always proceeded to remove them by causing a state of unconsciousness. It would of course be wrong to choose a moment for beginning the inhalation, when fear was producing a very marked depression of the circulation. On feeling the pulse of a gentleman, about twenty-one years of age, in March 1855, who had just seated himself in the chair to take chloroform, previous to having some teeth extracted, by Mr. Thos. A. Rogers, I found it to be small, weak, and intermitting, and it became more feeble as I was feeling it. I told the patient that he would feel no pain, and that he had nothing whatever to apprehend. His pulse immediately improved. He inhaled the chloroform, had his teeth extracted, woke up, and recovered without any feeling of depression. Now if the inhalation had been commenced in this case, without inquiry or explanation, the syncope which seemed approaching would probably have taken place, and it would have had the appearance of being caused by the chloroform, although not so in reality.
MODE OF ADMINISTERING CHLOROFORM.
The experiments previously related show that air containing rather less than two grains of chloroform, in one hundred cubic inches, is capable of causing a state of insensibility, sufficiently deep for surgical operations; but in a creature the size of the human being, an inconvenient length of time would be occupied in causing insensibility with vapour so much diluted. About four cubic inches of vapour, or rather more than five grains of chloroform to each hundred cubic inches of air, is the proportion which I have found most suitable in practice for causing insensibility to surgical operations. In medical and obstetric cases, it should be inhaled in a more diluted form.
Dr. Simpson recommended chloroform to be administered on a handkerchief—the method in which sulphuric ether was administered by Dr. Morton, in the first case in which he exhibited that medicine. The objection to giving chloroform on a handkerchief, especially in surgical operations, where it is necessary to cause insensibility, is that the proportions of vapour and of air which the patient breathes cannot be properly regulated. Indeed, the advocates of this plan proceed on the supposition that there is no occasion to regulate these proportions, and that it is only requisite that the patient should have sufficient air for the purposes of respiration, and sufficient chloroform to induce insensibility, and all will be right.[53] The truth is, however, that if there be too much vapour of chloroform in the air the patient breathes, it may cause sudden death, even without previous insensibility, and whilst the blood in the lungs is of a florid colour. Chloroform may indeed be inhaled freely from a handkerchief without danger, when it is diluted with one or two parts by measure of spirits of wine, but the chloroform evaporates in largest quantity at first, and less afterwards, until a portion of the spirit is left behind by itself. The process, however, of inhaling chloroform from a handkerchief is always uncertain and irregular, and is apt to confirm the belief in peculiarities of constitution, idiosyncrasies and predispositions, which have no existence in the patient.[54]
The most exact way in which it is practicable to exhibit chloroform to a patient about to undergo an operation, is to introduce a measured quantity into a bag or balloon of known size, then to fill it up by means of the bellows, and allow the patient to inhale from it; the expired air being prevented from returning into the balloon, by one of the valves of the face-piece to which it is attached. I tried this plan in a few cases, in 1849, with so much chloroform in the balloon as produced four per cent. of vapour in proportion to the air. The effects were extremely uniform, the patients becoming insensible in three or four minutes, according to the greater or less freedom of respiration; and the vapour was easily breathed, owing to its being so equally mixed with the air. I did not try, however, to introduce this plan into general use, as the balloon would sometimes have been in the way of the surgeon, and filling it with the bellows would have occasioned a little trouble. It seemed necessary to sacrifice a little of absolute perfection to convenience, and I therefore continued the plan which I had already followed.
The great point to be observed in causing insensibility by any narcotic vapour, is to present to the patient such a mixture of vapour and air as will produce its effects gradually, and enable the medical man to stop at the right moment. Insensibility is not caused so much by giving a dose as by performing a process. Nature supplies but one mixture of diluted oxygen, from which each creature draws as much as it requires, and so, in causing narcotism by inhalation, if a proper mixture of air and vapour is supplied, each patient will gradually inhale the requisite quantity of the latter to cause insensibility, according to his size and strength. It is indeed desirable to vary the proportions of vapour and air, but rather according to the purpose one has in view, whether medicinal, obstetric, or surgical, than on account of the age or strength of the patient; for the respiratory process bears such a relation to the latter circumstances, as to cause each person to draw his own proper dose from a similar atmosphere in a suitable time.
The inhaler represented in the adjoining engraving is, with some slight alterations, the same that I have employed since the latter part of 1847. It is made of metal, and consists of a double cylinder, the outer space of which contains cold water, and the inner serves for the evaporation of the chloroform which the patient is to breathe. Into
the inner part of the cylinder there is screwed a frame, having numerous openings for the admission of air, and four stout wires which descend nearly to the bottom of the space, and are intended to support two coils of stout bibulous paper, which are tied round them, and reach to the bottom of the inhaler. In the lower part of this paper four notches are cut, to allow the air to pass in the direction indicated by the arrows. As the quantity of chloroform which is put in should never fill the apertures or notches, the air which passes through the inhaler meets with no obstruction whatever. There is a glass tube communicating with the interior of the inhaler, and passing to the outside, to enable the operator to see when the chloroform requires to be renewed. The elastic tube which connects the inhaler to the face-piece is three-quarters of an inch in internal diameter, to allow of the passage of as much air as the patient can possibly breathe. On the introduction of the practice of inhaling sulphuric ether there was no tubing in this country fit to be breathed through; that in ordinary use was only about one-third, or three-eighths of an inch in diameter,—not more than a quarter of the proper calibre.
The face-piece, to include both the mouth and nostrils, of which that shown in the engraving is one of the modifications, is one of the greatest mechanical aids to the process of inhalation which has been contrived in modern times. Dr. Francis Sibson is its inventor. Dr. Hawkesley did indeed contrive a very similar one about the same time as Dr. Sibson,—early in 1847,—but he did not make it known. Dr. Ingen Housz made patients inhale oxygen through the nostrils by means of a bottle of India rubber with the bottom cut off; and Mr. Waugh, of Regent Street, had more recently contrived a mouth-piece to be adapted outside the lips, but the usual practice of inhalation previous to 1847, was for the patient to draw in the medicated air by means of a tube placed in the mouth. This led generally to great awkwardness at first, as the patient usually began to puff as if he were smoking a pipe; and it had the further inconvenience, in the administration of ether, that the tube dropped from the mouth, and the patient began to breathe by the nostrils, just as he was getting unconscious. The sides of the face-piece delineated in the engraving are made of thin sheet lead, which is pliable, and enables it to be adapted exactly to the inequalities of the face, and the patient can breathe either by the nostrils or mouth, just as his will, or instinct, or other nervous functions, determine.
I have introduced two valves into this face-piece, one which rises on inspiration, to admit the air and vapour from the inhaler, and closes again on expiration, and the other which rises to allow the expired air to escape. I contrived the latter valve to turn more or less to one side, as indicated by the additional line in the engraving, and thus admit more or less of the external air to dilute still further that which has passed through the inhaler, and become charged with vapour. By this means the patient can begin by breathing air containing very little vapour, and more and more of the air which has passed over the moistened bibulous paper can be admitted, as the air-passages become blunted to the pungency of the vapour.
The object of the water-bath is to supply the caloric which is rendered latent, and carried off, as the chloroform is converted into vapour, and thus to render the process of inhalation steady and uniform. Without the water-bath, the evaporation of the chloroform would soon reduce the temperature of the inhaler below the freezing point of water, and limit very much the amount of vapour the patient
would inhale; and if the apparatus were warmed by the hand, the temperature would be too high, and the amount of vapour too great. A medical author of great reputation in Paris sent to inquire at what temperature I used the water-bath, and being informed, at the ordinary temperature, published his opinion that it had no effect, and might as well be left off. He appeared not to have considered the relations of heat, either to liquids or vapours.
In arranging the bibulous paper in the inhaler, it is my object to contrive that the air passing through, in the ordinary process of inhalation, and at the ordinary temperature of about 60°, shall take up about five per cent. of vapour. This quantity can be diminished, as much as is desired, by turning the expiratory valve of the face-piece a little to one side; and in winter I usually place a short coil of bibulous paper against the outer circumference of the inside of the inhaler, in addition to the central coils which are delineated.
I commonly put two, or two and a half, fluid drachms of chloroform into the inhaler at first. About a drachm of this is absorbed by the filtering paper, and the rest remains at the bottom of the inhaler; and in a protracted operation, when it is seen, by means of the glass tube, that the latter part of the chloroform has disappeared, more is added, by a drachm or so at a time, to prevent the paper ever becoming dry. Mr. Matthews, 8, Portugal Street, Lincoln’s Inn Fields, makes the inhaler. There are smaller face-pieces for children. The patient never inhales in so upright a posture as the artist has represented.
There are several other kinds of apparatus in use for the inhalation of chloroform. The most usual consist of Dr. Sibson’s face-piece more or less altered, and with a small piece of sponge placed inside. The apparatus which is in most reputation on the continent is that of M. Charrière; it consists of a glass vase with suitable valves, and a fabric for exposing a surface wetted with chloroform to the air which passes through it.
M. Duroy, of Paris, has contrived an ingenious, but very complicated, apparatus, which he calls an anæsthesimeter. The object of it is to regulate the amount of chloroform which is inhaled in a given time, and this can be varied from four to sixty drops in the minute; but the experiments which I have related show that the quantity of chloroform employed is not so important as the
proportion of it in the inspired air; and although each of these circumstances has considerable influence over the other, in many cases there are conditions in which no regular relation exists between them. For instance, if the inhaler were supplied with sixty drops of chloroform per minute, these sixty drops weigh twenty grains, and produce 15·3 cubic inches of vapour; and if an adult patient were breathing the average quantity of four hundred cubic inches per minute, the air he would breathe would contain nearly four per cent. of vapour, which would answer extremely well; but if the breathing were slow or feeble, or if he should hold his breath for an interval and commence again, he might breathe air much more highly charged with vapour. Indeed it would depend on the amount of surface moistened with chloroform, the temperature of the air, and other physical conditions, whether or not the air he inhaled might not be charged with chloroform to a dangerous degree; whilst, on the other hand, if the breathing were deep and rapid, as often happens whilst the patient is getting slightly under the influence of the chloroform;—if, for instance, the patient were to breathe at the rate of 1,600 cubic inches, instead of 400, the air he would inhale would contain less than one per cent. of vapour, and he would not become insensible with the utmost supply of the anæsthesimeter, till his breathing should be moderated. M. Duroy also follows the rude and objectionable plan of using a nose clasp, and thus compelling the patient to breathe by the mouth alone.
It is advisable to request the patient to breathe gently and quietly, when he commences to inhale chloroform; in other words, to do nothing but conduct himself as if he were about to fall asleep naturally; for, if he breathes deeply, the vapour feels much more pungent than it otherwise would do, and is apt to excite coughing, or a feeling of suffocation.
In using the inhaler described above, the patient should commence to inhale with the expiratory valve of the face-piece turned on one side, and it should be gradually advanced to the required extent, over the opening it is intended to cover, as the sensibility of the lungs becomes diminished. Not only patients with phthisis or bronchitis, but many sensitive and irritable persons with sound lungs, have a great intolerance of the vapour of chloroform at first, on account of its pungency; and it is necessary to occupy two or three minutes in
gradually commencing the inhalation, before the patient makes any appreciable progress towards insensibility. In administering chloroform to children also, it is desirable to begin very gently; by this means, and with a little persuasion, one generally succeeds in getting them to inhale voluntarily; although, occasionally, it is necessary to use a little force to accomplish one’s purpose.
In certain cases of the medicinal application of chloroform, and also in obstetric cases, where the pains are not severe, it is unnecessary to render the patient unconscious, but for surgical operations this is nearly always requisite. No force should ever be employed so long as the patient is conscious, unless it be to children or lunatics; but some patients become excited as soon as they are unconscious, and attempt to leave the couch, or push away the chloroform; under such circumstances, if they cannot be calmed by what is said to them, they should be held, and the vapour should be steadily and gently continued, for a minute or two, till a state of quietude is produced. By far the greater number of patients remain quiet as they become unconscious, but there is no difficulty in ascertaining whether a patient is unconscious or not. If the eyelids remain open, the countenance shows whether the patient is conscious or not; and, if they are closed, it is only necessary to touch them gently, to ascertain this circumstance. If he is still conscious, he will look at his medical man, and probably speak, or, at all events, show intelligence in his countenance.
Signs of Insensibility. The absence of consciousness, and a state of quietude, are both requisite before the commencement of a surgical operation, and they go a good way towards the preparation of a patient for it, but these symptoms may be present and the patient not be ready for an operation. The surgeon wishes to know whether he will lie still under the knife, or whether he will make a resistance and outcry which he would probably not make in his waking state. Some surgeons have recommended that the patient should be pricked with the point of a knife or some other instrument. This is not a good or satisfactory plan, however, for a person will often show no sign of feeling a slight prick, when a severe incision would rouse him to resistance. A more elegant and successful plan is to raise the eyelid gently, by placing one finger just below the eyebrow, and then to touch its ciliary border very lightly with another finger. This should
not be done roughly nor too frequently, for fear of exhausting the sensibility when it is slight. Just after unconsciousness is induced, the eyelids are often closed very strongly when their margins are touched, especially in females, and there seems to be a positive hyperæsthesia; this, however, is only apparent, and arises from the control of the will being removed, whilst sensibility remains. By continuing the chloroform, the sensibility of the edges of the eyelids diminishes until, at last, they may be touched without causing winking. Under these circumstances, the most severe operation may, in almost every case, be commenced without sign of pain. I have employed this test of the sensibility or insensibility of the patient ever since chloroform has been in use, and also in the employment of ether in 1847, and I am satisfied that it affords more reliable information on this point than any other single symptom. It even indicates the amount of sensibility where a little remains; when, for instance, touching the margin of the eyelids causes very slight and languid winking, the patient will commonly flinch a little if the knife is used, but only in a manner that can be easily restrained, and will not interfere with the majority of operations. The cases, in which the indication afforded by the eyelids is not always to be depended on, are those of hysterical patients, in whom there is sometimes no winking on touching the eyelids, even when unconsciousness is scarcely induced. In such cases, one must judge by the other symptoms, and also by the length of time the patient has inhaled, the strength of the vapour, and depth and activity of the breathing. Indeed, these conditions should be observed and taken into account in every case; and all the symptoms exhibited by the patient should be watched, such as the expression of the face, the state of the breathing, and the condition of the limbs with regard to their tension or relaxation. The last is indeed sometimes relied on as the chief or sole sign whether the operation may be commenced, but it is of itself very insufficient, and even fallacious. The patient may allow his limbs to lie relaxed when he is scarcely unconscious, and not at all insensible, merely because he is not exerting his will upon them; if the arm is lifted, it may fall listlessly down again, at a time when the knife of the surgeon would rouse the patient to active resistance. Indeed, the limbs, which have been lying relaxed, may become tense as the effect of the chloroform increases, and may remain so during a short operation, in which there is no sign of pain.
Although the pulse of itself gives no indication as to how far a patient is under the influence of chloroform, it is proper to pay attention to it, not only during the first administration of the chloroform, but also throughout the operation, especially if it be attended with much bleeding. The pulse sometimes becomes intermittent or irregular during the administration of chloroform, more especially in elderly persons. This more commonly happens in the first exhibition of it, than when it is repeated during the course of an operation. I have not seen any harm from either of these conditions, but it is well to intermit the chloroform for ten or fifteen seconds, and let the patient have a few inspirations without it, if the pulse is not in a satisfactory condition. If the precaution be taken to ensure that the air the patient breathes shall never contain more than five per cent. of vapour, the pulse can never be seriously affected by the direct action of the chloroform, and the state of the breathing affords the best warning against continuing the inhalation too long at a time.
The breathing is fortunately also a sign that cannot be overlooked. It is by the breath that the chloroform enters, and it is extremely improbable that any one would go on giving the vapour after the breathing became stertorous and laboured.
The patient sometimes holds his breath after he is unconscious, and before he is insensible; this occurs under two conditions: first, after deep and rapid breathing, during which the patient seems to absorb more oxygen than is immediately required, under the circumstances; and in this condition, I have known him hold his breath for a whole minute, whilst the pulse was unaffected. The other condition in which the breathing is suspended, is when there is rigidity in the third degree of narcotism, and the respiratory muscles seem to partake of the general rigidity; the holding of the breath in these cases seldom continues so long as under the former circumstances. I do not consider that there is any danger from either of these kinds of suspension of breathing. I believe it always returns as soon as there is a want of oxygen in the system. Of course the inhaler need not be applied to the face when the patient is not breathing, and he may as well have an inspiration or two without chloroform when the breathing is renewed. It is seldom requisite to carry the effects of chloroform so far that the breathing becomes
stertorous, and whenever stertor is observed, the inhalation should be suspended; under these circumstances, the patient is always insensible. In some cases, in which a little more chloroform has been inhaled than is necessary, the patient breathes for half a minute or so by the diaphragm only, and breathes in fact hardly at all. The abdomen rises and falls freely, but, from the muscles of the chest not fixing the ribs, hardly any air enters the lungs, and the face becomes rather livid; meanwhile the pulse goes on very well, and at last the patient draws a deep, sonorous inspiration, the face resumes its proper colour, and all is right again. I have not heard of any accident from chloroform commencing in this manner. This state of breathing, when it does occur, usually takes place a few seconds after the inhalation has been left off, and arises from the accumulation of the effects of the chloroform, owing to the absorption into the system of the vapour which was contained in the lungs at the time when the inhalation was discontinued.[55] This accumulation or increase of the effects of chloroform lasts for about twenty seconds; it is not dangerous unless the vapour is inhaled of too great strength, but it should be borne in mind in all cases. It may be prevented altogether, by reducing the strength of the vapour, just as the patient is getting insensible, or by giving it with intermissions of a few seconds, at this time.
The rigidity and struggling previously mentioned (pages 39 and 50) as occurring occasionally in the third degree of narcotism, more particularly in robust persons, often form a very prominent feature in the effects of chloroform; and have sometimes caused the medical man to discontinue the exhibition of chloroform, under the belief that it did not agree with the constitution of the patient, and that its further exhibition would be unsafe. The proper course to pursue is to continue the inhalation gently, till the struggling and rigidity are subdued. The patient is often insensible before these symptoms are subdued, but it is necessary to have him quiet, in order to enable the surgeon to operate with convenience and safety. I have always succeeded in subduing the involuntary struggling and rigidity, but have occasionally occupied five or six minutes in doing so. It is desirable to proceed slowly and cautiously, because, when these symptoms occur, the patient has already absorbed nearly the usual quantity of chloroform, and he often holds his breath, and then takes
a sudden and deep inspiration, when he might inhale an overdose of vapour, unless it were presented to him in a well diluted state.
When the rigidity and struggling are subdued, the breathing, in some cases, becomes stertorous, and relaxation of the muscles takes place, the limbs appearing quite flaccid; but by proceeding gently, these effects may generally be avoided, and the patient becomes quiet, whilst the breathing is natural, and the muscles are in a moderate state of tension. If the operator should be afraid to proceed with the exhibition of chloroform, on account of the violence of the muscular spasm and rigidity, it will be satisfactory to him to know that, if the inhalation is resumed in a few minutes, these symptoms will be less violent than at first.
Struggling and rigidity are less likely to occur, when chloroform is administered slowly, than under opposite circumstances; but it is impossible to prevent these phenomena altogether in certain patients. After they are once subdued, they but very rarely recur during the operation; the patient, in most cases, seems to take on, when he is subdued by the chloroform, the same relation to it that women, children, and persons in a state of debility have from the first. M. Chassaignac has called this condition one of tolerance of the chloroform. It is a condition in which the patient bears both the chloroform and the operation very comfortably; but tolerance of a medicine is generally meant to imply that the patient can take it in larger quantity than before. But this is the reverse of what occurs when the patient is in a tranquil state from chloroform; he has already absorbed a considerable quantity, which has most likely penetrated deeply into the tissues, and he certainly does not require, and could not bear, so much as in the earlier stage of inhalation, where he is restless and breathing more quickly, and thus exhaling and getting rid of the chloroform at a greater rate.
It might be a question whether the absence of muscular excitement, in a number of cases, does not arise from the circumstance that anæsthesia, or absence of common sensibility, is obtained, and the operation performed, at a stage of narcotism anterior to that in which the muscular rigidity and spasm occur. This is true in a few cases, but I am satisfied by careful observation that, in the greater number of instances in which muscular excitement is absent, it would not occur at all, though the inhalation should be
